If [math]X[/math] has a normal distribution with mean 2 and standard deviation 1.5, determine [math]\operatorname{P}[X^2 - X -2 \gt 0 ] [/math].

1. 0
2. 0.4772
3. 0.5
4. 0.5228
5. 1

A math teacher teaches calculus at a local community college. The teacher has split his students into three groups based on attendance levels: low attendance, medium attendance and high attendance. The upcoming final exam scores are assumed to be normally distributed for each student with the mean and standard deviation dependent on the attendance level of the student:

The passing score is set at the highest integer value that yields an expected pass rate for high attendance students of at least 90%. Determine the expected % of students that will fail the final exam.

1. 0.33
2. 0.3794
3. 0.4
4. 0.4294
5. 0.5

An actuary is using a normal approximation to model loss distributions. To calibrate the parameters of the normal distribution, the actuary uses historical loss data. You are given the following:

• The sample mean equals 500
• 10% of the historical losses are larger than 800

Approximate the 95^th percentile of the loss distribution.

1. 678.15
2. 750.33
3. 856.11
4. 884.96
5. 900.25

Losses without deductible (full coverage) has a uniform distribution on [0,b]. When a deductible equal to rb is in effect, the payment variance is 0.75(1-r)² times the payment variance without deductible. Determine which interval contains the value of r.

• [0.55,0.6]
• [0.68,0.73]
• [0.75,0.79]
• [0.82,0.85]
• [0.9,0.95]

A policy's full coverage claim severity is uniform on [0,2000] with a per-claim deductible of $500 and per-claim limit, applied before the deductible, of $1,500. A claim is reported when the loss to the insured exceeds the deductible. Determine the expected loss to the insurer per reported claim.

• 501.55
• 625.67
• 666.67
• 700.67
• 872.55

An insurance policy reimburses dental expense, [math]X[/math], up to a maximum benefit of 250. The probability density function for [math]X[/math] is:

where [math]c[/math] is a constant. Calculate the median benefit for this policy.

• 161
• 165
• 173
• 182
• 250

The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company expects that 30% of high-risk drivers will be involved in an accident during the first 50 days of a calendar year.

Calculate the portion of high-risk drivers are expected to be involved in an accident during the first 80 days of a calendar year.

• 0.15
• 0.34
• 0.43
• 0.57
• 0.66

The lifetime of a printer costing 200 is exponentially distributed with mean 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, a one-half refund if it fails during the second year, and no refund for failure after the second year.

Calculate the expected total amount of refunds from the sale of 100 printers.

• 6,321
• 7,358
• 7,869
• 10,256
• 12,642

The owner of an automobile insures it against damage by purchasing an insurance policy with a deductible of 250. In the event that the automobile is damaged, repair costs can be modeled by a uniform random variable on the interval (0, 1500).

Calculate the standard deviation of the insurance payment in the event that the automobile is damaged.

• 361
• 403
• 433
• 464
• 521

The time to failure of a component in an electronic device has an exponential distribution with a median of four hours.

Calculate the probability that the component will work without failing for at least five hours.

• 0.07
• 0.29
• 0.38
• 0.42
• 0.57